Let $m \in \mathbb N$. Find a necessary and sufficient condition for $m$ such that the ideal $(m,x^2+y^2)$ is prime in $\mathbb Z[x,y]$.
I have to find for which $m$ the quotient ring is an integral domain. I do not know how to use the isomorphism theorem: is it ture that $$ \mathbb Z[x,y] /(m,x^2+y^2) \cong (\mathbb Z_m[x])[y]/(x^2+y^2)? $$
What should we do? Thanks in advance.